On the uniqueness of (p,h)(p,h)-gonal automorphisms of Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$. A cyclic subgroup of prime order $p$ of $Aut(X)$ is called properly $(p,h)$-gonal if it has a fixed point and the quotient surface has genus $h$. We show that if $p>6h+6$, then a properly $(p,h)$-gonal subgroup of $Aut(X)$ is unique. We also discuss some related results.Comment: final version, 9 pages, minor improvements, added 2 reference

Metabelian Groups Acting on Compact Riemann Surfaces

Sin resume

On fixed points of automorphisms of non-orientable unbordered Klein surfaces

In 1973, Macbeath found a general formula for the number of points fixed by an arbitrary orientation preserving automorphism of a Riemann surface X. It was given in terms of a group G of conformal automorphisms of X and the ramification data of the covering X --> X/G, which corresponds to the so called universal covering transformation group. In these terms, for the case of a cyclic group of automorphisms of an unbordered non-orientable Klein surface, the formula was given later by Izquierdo and Singerman and here we find formulas valid for an arbitrary (finite) group G of automorphisms

Groups of automorphisms of cyclic trigonal Riemann surfaces

AbstractWe give the list of all groups G acting as a group of automorphisms of some cyclic trigonal compact Riemann surface X of genus g⩾5 and containing the trigonality automorphism group. An abstract group G may act in different ways producing coverings X→X/G with different ramification type; the list of all such different ramification types is also given

Symmetry types of hyperelliptic Riemann surfaces

Let $X$ be a compact hyperelliptic Riemann surface which admits anti-analytic involutions (also called symmetries or real structures). For instance, a complex projective plane curve of genus two, defined by an equation with real coefficients, gives rise to such a surface, and complex conjugation is such a symmetry. In this memoir, the real structures $\tau$ of $X$ are classified up to isomorphism (i.e., up to conjugation). This is done as follows: the number of connected components of the set of fixed points of $\tau$ together with the connectedness or disconnectedness of the complementary set in $X$ classifies $\tau$ topologically; they determine the species of $\tau$, which only depends on the conjugacy class of $\tau$ (however, different conjugacy classes may have identical species). On these grounds, for a given genus $g\ge2$, the authors first give a list of all full groups of analytic and anti-analytic automorphisms of genus $g$ compact hyperelliptic Riemann surfaces. For every such group $G$, the authors compute polynomial equations for a surface $X$ having $G$ as full group and then find the number of conjugacy classes containing symmetries; they also compute a representative $\tau$ in every such class. Finally, they compute the species corresponding to such classes. This memoir is an exhaustive piece of work, going through a case-by-case analysis. The problem for general compact Riemann surfaces dates back to 1893, when {\it F. Klein} [Math. Ann. 42, 1--29 (1893)] first studied it. For zero genus, it is easy. For genus one, that is, for elliptic surfaces, it was solved by {\it N. Alling} ["Real elliptic curves" (1981)]. Partial results for hyperelliptic surfaces of genus two were obtained by {\it E. Bujalance} and {\it D. Singerman} [Proc. Lond. Math. Soc. 51, 501--519 (1985)]

CO observations of symbiotic stellar systems

We have performed mm-wave observations with the IRAM 30m telescope of the 12CO J=2-1 and J=1-0, 13CO J=2-1 and J=1-0, and SiO J=5-4 transitions in the symbiotic stars R Aqr, CH Cyg, and HM Sge. The data were analyzed by means of a simple analytical description of the general properties of molecular emission from the inner shells around the cool star. Numerical calculations of the expected line profiles, taking into account the level population and radiative transfer under such conditions, were also performed. Weak emission of 12CO J=2-1 and J=1-0 was detected in R Aqr and CH Cyg; a good line profile of 12CO J=2-1 in R Aqr was obtained. The intensities and profile shapes of the detected lines are compatible with emission coming from a very small shell around the Mira-type star, with a radius comparable to or slightly smaller than the distance to the hot dwarf companion, 10$^{14}$ - 2 10$^{14}$ cm. We argue that other possible explanations are improbable. This region probably shows properties similar to those characteristic of the inner shells around standard AGB stars: outwards expansion at about 5 - 25 km/s, with a significant acceleration of the gas, temperatures decreasing with radius between about 1000 and 500 K, and densities ~ 10$^9$ - 3 10$^8$ cm$^{-3}$. Our model calculations are able to explain the asymmetric line shape observed in 12CO J=2-1 from R Aqr, in which the relatively weaker red part of the profile would result from selfabsorption by the outer layers (in the presence of a velocity increase and a temperature decrease with radius). The mass-loss rates are somewhat larger than in standard AGB stars, as often happens for symbiotic systems. In R Aqr, we find that the total mass of the CO emitting region is ~ 2 - 3 10$^{-5}$ Mo, corresponding to M' ~ 5 10$^{-6}$ - 10$^{-5}$ Mo/yr, and compatible with results obtained from dust emission. Taking into account other existing data on molecular emission, we suggest that the small extent of the molecule-rich gas in symbiotic systems is mainly due to molecule photodissociation by the radiation of the hot dwarf star.Comment: 11 pages, 4 figure